Morbid curiosity about "more better"

However grotesque the "more better" construction may be, I think this sentence is grammatically correct.

When diagrammed, more does not modify better — it modifies the object, things —, so perhaps you have found a case where "more" + "better" can be used correctly?

It is certainly possible to have more and better things to do. The wording as you have it may provoke comment (and in some circles, censure), so you might want to try to stick an and in between the two words

I have so many more and better things to do than grade your homework.

You could also emphasize the distinct nature of the words with a comma:

I have so many more, better things to do than grade your homework.

However, if you really only mean you have other things to do that are better, it's simpler just to say

I have so many better things to do than grade your homework.

Note that in some dialects the "more" in "more better" just acts as an intensifier and is unobjectionable. See Spike Lee's Mo' Better Blues as a case in point.

Note also that if you do include more in addition to better you may be implying that grading that person's homework is included in the things you have to do.

It's certainly true that in some dialectal usages, more can be used as intensifier for better - I know at least one person who playfully amplifies the form as "Ah! I feel much more betterer now!"

But even sticking to standard English, there's nothing at all wrong with OP's example. As @horatio comments, you can have "one better thing", so you can obviously have "more better things".

So OP's student is quite correct - "better things" can indeed be parsed as a "single lexical item" capable of being modified by "more". The fact that it would be dialectal/non-standard to say...

"?*I have a more better thing to do than grade your homework."

...is effectively a red herring which has no real bearing on the grammaticality of the example.

There's the subtle issue of whether or not grading the addressee's homework is included within the scope of "better things", but OP provides insufficient context to rule on this. Preceded by...

"Prioritising is always an issue for me when I have a lot to do, and I often can't make a choice and get started. Tonight I'll probably fail to make a decision, veg out with the TV, and end up feeling guilty about wasting my time again."

...the implication would be that grading is a "better thing" than watching TV, and the teacher also has many other things to do which are also better than watching TV. But with...

"I do have a life outside school, you know. And they don't pay me for extracurricular activities like preparing lesson plans and monitoring students' progress."

...it's safe to assume grading homework isn't one of the "better things" the teacher has in mind.

It would sound more natural to use other rather than more here, as the contrast with better is awkward. A comma is still needed because other and better don't catenate smoothly; the construction parallels the first, not the second, example below:

I have many other, green socks. (I have many other socks - they are green.)

I have many other green socks. (I have many other green socks besides these examples.)

A contrived but grammatically correct example using so many more better things is:

Your language tutor has kindly told me she will grade your homework, even though it is a week late and probably as poorly executed as usual. She kindly relieved me of the task - she has many better things to do, but I have so many more better things to do this week that she said she'd help me out. Next time, you won't get a grade.

It is 100% grammatically correct, but 100% awkward-sounding without sufficient context. In one sense, "More better" in and of itself--"more" being an adverb modifying "better"--is grammatically incorrect. In another sense, considering that "better" is just an irregular way of saying what would be "more good" or "gooder", "more better" is just redundant.

But, "more" can also mean more in number, i.e. additional better things. In this sense it is very much grammatically correct. Consider the following conversation (argument):

I have better things to do than this.
Oh yea? Well I have even more better things to do than you do. So there!

The silliness of such an argument notwithstanding, with the extra context given by the conversation, and extra "sugar" added to the sentence (i.e. "even" and "than you"), it sounds a lot more natural. If all of this were removed, we'd be left with

I have more better things to do.

So the conclusion: the expression is syntactically ambiguous and can be parsed in three different ways--two of those three are grammatically incorrect. Two out of three makes for a grammatically correct, but very awkward sounding expression.